When CBS Corp. Chief Executive Leslie Moonves, Viacom Inc. Chief Executive Bob Bakish and their top lieutenants met in January over deli sandwiches at CBS’s Los Angeles offices to discuss merging their companies, things started out cordially.

The two sides agreed on the strategic merits of a tie-up, according to people familiar with the matter. But as the meal wore on, it became clear that the fate of the deal would turn on one question: What about Bob? That is, would Mr. Moonves accept Mr. Bakish as his No. 2 and likely successor...